Sharp bounds ξ(C) ≤ τ(C) for stochastically increasing copulas and ξ(C) ≤ ρ(C) under LTD and RTI are proved, with equality cases identified via ordinal sums and checkerboard examples.
Measures of association for approximating copulas
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abstract
This paper studies closed-form expressions for multiple association measures of copulas commonly used for approximation purposes, including Bernstein, shuffle--of--min, checkerboard and check--min copulas. In particular, closed-form expressions are provided for the recently popularized Chatterjee's $\xi$, which quantifies the dependence between two random variables. Given an absolutely continuous bivariate copula $C$ with TP$_2$ density and approximating $n\times n$-checkerboard copula $C_n$, we show that $\xi(C_n) \le \xi(C)$ with $\xi(C_n) \to \xi(C)$ as $n\to\infty$.
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Kendall and Spearman bounds for Chatterjee's rank correlation under positive dependence
Sharp bounds ξ(C) ≤ τ(C) for stochastically increasing copulas and ξ(C) ≤ ρ(C) under LTD and RTI are proved, with equality cases identified via ordinal sums and checkerboard examples.